This application claims the priorities of Japanese Patent Application No. 2000-277444 filed on Sep. 13, 2000 and Japanese Patent Application No. 2001-022633 filed on Jan. 31, 2001, which are incorporated herein by reference.
1. Field of the Invention
The present invention relates to a fringe analysis error detection method and fringe analysis error correction method using Fourier transform method when analyzing fringe images by using phase shift methods; and, in particular, to a fringe analysis error detection method and fringe analysis error correction method in which PZTs (piezoelectric elements) are used for shifting the phase, and Fourier transform method is utilized when analyzing thus obtained image data with fringe patterns such as interference fringes, whereby the analyzed value can be made more accurate.
2. Description of the Prior Art
While light wave interferometry, for example, has conventionally been knows as an important technique concerning precise measurement of a wavefront of an object, there have recently been urgently demanded for developing an interferometry technique (sub-fringe interferometry) which can read out information from a fraction of a single interference fringe (one fringe) or less due to the necessity of measuring a surface or wavefront aberration at an accuracy of {fraction (1/10)} wavelength or higher.
Known as a typical technique widely used in practice as such a sub-fringe interferometry technique is the phase shift fringe analyzing method (also known as fringe scanning method or phase scanning method) disclosed in xe2x80x9cPhase-Measurement Interferometry Techniques,xe2x80x9d Progress in Optics, Vol. XXVI (1988), pp. 349-393.
In the phase shift method, one or more phase shift element such as PZTs (piezoelectric element), for example, are used for phase-shifting the relative displacement between an object to be observed and the reference, interference fringe images are captured each time when a predetermined phase amount is shifted, the interference fringe intensity at each point on the surface to be inspected is measured, and the phase of each point on the surface is determined by using the result of measurement.
For example, when carrying out a four-step phase shift method, respective interference fringe intensities I1, I2, I3, I4 at the individual phase shift steps are expressed as follows:                                                         I              1                        ⁡                          (                              x                ,                y                            )                                =                                                                      I                  0                                ⁡                                  (                                      x                    ,                    y                                    )                                            ⁡                              [                                  1                  +                                      γ                    ⁡                                          (                                              x                        ,                        y                                            )                                                                      ]                                      ⁢                          xe2x80x83                        ⁢                          cos              ⁡                              [                                  φ                  ⁡                                      (                                          x                      ,                      y                                        )                                                  ]                                                    ⁢                  
                ⁢                                            I              2                        ⁡                          (                              x                ,                y                            )                                =                                                                      I                  0                                ⁡                                  (                                      x                    ,                    y                                    )                                            ⁡                              [                                  1                  +                                      γ                    ⁡                                          (                                              x                        ,                        y                                            )                                                                      ]                                      ⁢                          xe2x80x83                        ⁢                          cos              ⁡                              [                                                      φ                    ⁡                                          (                                              x                        ,                        y                                            )                                                        +                                      π                    /                    2                                                  ]                                                    ⁢                  
                ⁢                                            I              3                        ⁡                          (                              x                ,                y                            )                                =                                                                      I                  0                                ⁡                                  (                                      x                    ,                    y                                    )                                            ⁡                              [                                  1                  +                                      γ                    ⁡                                          (                                              x                        ,                        y                                            )                                                                      ]                                      ⁢                          xe2x80x83                        ⁢                          cos              ⁡                              [                                                      φ                    ⁡                                          (                                              x                        ,                        y                                            )                                                        +                  π                                ]                                                    ⁢                  
                ⁢                                            I              4                        ⁡                          (                              x                ,                y                            )                                =                                                                      I                  0                                ⁡                                  (                                      x                    ,                    y                                    )                                            ⁡                              [                                  1                  +                                      γ                    ⁡                                          (                                              x                        ,                        y                                            )                                                                      ]                                      ⁢                          xe2x80x83                        ⁢                          cos              ⁡                              [                                                      φ                    ⁡                                          (                                              x                        ,                        y                                            )                                                        +                                      3                    ⁢                                          π                      /                      2                                                                      ]                                                                        (        1        )            
where
x and y are coordinates;
xcfx86(x, y) is the phase;
I0(x, y) is the average optical intensity at each point; and
xcex3(x, y) is the modulation of interference fringes.
From these expressions, the phase xcfx86(x, y) can be determined and expressed as:                               φ          ⁡                      (                          x              ,              y                        )                          =                              tan                          -              1                                ⁢                      xe2x80x83                    ⁢                                                                      I                  4                                ⁡                                  (                                      x                    ,                    y                                    )                                            -                                                I                  2                                ⁡                                  (                                      x                    ,                    y                                    )                                                                                                      I                  1                                ⁡                                  (                                      x                    ,                    y                                    )                                            -                                                I                  3                                ⁡                                  (                                      x                    ,                    y                                    )                                                                                        (        2        )            
Though the phase shift methods enable measurement with a very high accuracy if the predetermined step amount can be shifted correctly, it may be problematic in that errors in measurement occur due to errors in the step amount and in that it is likely to be influenced by the disturbance during measurement since it necessitates a plurality of interference fringe image data items.
For sub-fringe interferometry other than the phase shift method, attention has been paid to techniques using the Fourier transform method as described in xe2x80x9cBasics of Sub-fringe Interferometry,xe2x80x9d Kogaku, Vol. 13, No. 1 (February, 1984), pp. 55 to 65, for example.
The Fourier transform fringe analysis method is a technique in which a carrier frequency (caused by a relative inclination between an object surface to be observed and a reference surface) is introduced, so as to make it possible to determine the phase of the object with a high accuracy from a single fringe image. When the carrier frequency is introduced, without consideration of the initial phase of the object, the interference fringe intensity i(x, y) is represented by the following expression (3):
i(x,y)=a(x,y)+b(x,y)cos[2xcfx80fxx+2xcfx80fyy+"PHgr"(x,y)]xe2x80x83xe2x80x83(3) 
where
a(x, y) is the background of interference fringes;
b(x, y) is the visibility of fringes;
xcfx86(x, y) is the phase of the object to be observed; and
fx and fy are carrier frequencies in the x and y directions respectively expressed by:             f      x        =                            2          ·          tan                ⁢                  xe2x80x83                ⁢                  θ          x                    λ        ,            f      y        =                            2          ·          tan                ⁢                  xe2x80x83                ⁢                  θ          y                    λ      
where xcex is the wavelength of light, and xcex8x and xcex8y are the respective inclinations of the object in the x and y directions.
The above-mentioned expression (3) can be rewritten as the following expression (4):
i(x,y)=a(x,y)+c(x,y)exp[i(2xcfx80fxx+2xcfx80fyy)]+c*(x,y)exp[i(2xcfx80fxx+2xcfx80fyy)]xe2x80x83xe2x80x83(4) 
where c(x, y) is the complex amplitude of the interference fringes, and c*(x, y) is the complex conjugate of c(x, y).
Here, c(x, y) is represented as the following expression (5):                               c          ⁡                      (                          x              ,              y                        )                          =                                            b              ⁡                              (                                  x                  ,                  y                                )                                      ⁢                          xe2x80x83                        ⁢                          exp              ⁡                              [                                  ⅈ                  ⁢                                      xe2x80x83                                    ⁢                                      φ                    ⁡                                          (                                              x                        ,                        y                                            )                                                                      ]                                              2                                    (        5        )            
The Fourier transform of expression (4) gives:
I(xcex7,xcex6)=A(xcex7,xcex6)+C(xcex7xe2x88x92fx,xcex6xe2x88x92fy)+C*(xcex7xe2x88x92fx,xcex6xe2x88x92fy)xe2x80x83xe2x80x83(6) 
where A(xcex7, xcex6) is the Fourier transform of a(x, y), and C(xcex7xe2x88x92fx, xcex6xe2x88x92fy) and C*(xcex7xe2x88x92fx, xcex6xe2x88x92fy) are the Fourier transforms of c(x, y) and c*(x, y), respectively.
Subsequently, C(xcex7xe2x88x92fx, xcex6xe2x88x92fy) is taken out by filtering, the peak of the spectrum positioned at coordinates (fx, fy) is transferred to the origin of a Fourier frequency coordinate system (also referred to as Fourier spectra plane coordinate system; see FIG. 6), and the carrier frequencies are eliminated. Then, inverse Fourier transform is carried out, so as to determine c(x, y), and the wrapped measured phase xcfx86(x, y) can be obtained by the following expression (7):                               φ          ⁡                      (                          x              ,              y                        )                          =                              tan                          -              1                                ⁢                                    Im              ⁡                              [                                  c                  ⁡                                      (                                          x                      ,                      y                                        )                                                  ]                                                    Re              ⁡                              [                                  c                  ⁡                                      (                                          x                      ,                      y                                        )                                                  ]                                                                        (        7        )            
where Im[c(x, y)] is the imaginary part of c(x, y), whereas Re[c(x, y)] is the real part of c(x, y).
Finally, unwrapping processing is carried out, so as to determine the phase "PHgr"(x, y) of the object to be measured.
In the Fourier transform fringe analyzing method explained in the foregoing, the fringe image data modulated by carrier frequencies is subjected to a Fourier transform method as mentioned above.
As mentioned above, the phase shift method captures and analyzes the brightness of images while applying a phase difference between the object light of an interferometer and the reference light by a phase angle obtained when 2xcfx80 is divided by an integer in general, and thus can theoretically realize highly accurate phase analysis.
For securing highly accurate phase analysis, however, it is necessary to shift the relative displacement between the sample and the reference with a high accuracy by predetermined phase amounts. When carrying out the phase shift method by physically moving the reference surface or the similar by using phase shift elements, e.g., PZTs (piezoelectric elements), it is necessary to control the amount of displacement of PZTs (piezoelectric elements) with a high accuracy. However, errors in displacement of the phase shift elements or errors in inclination of the reference surface or sample surface are hard to eliminate completely. Controlling the amount of phase shift or amount of inclination is actually a difficult operation. Therefore, in order to obtain favorable results, it is important to detect the above-mentioned errors resulting from the phase shift elements, and correct them according to thus detected values when carrying out the fringe analysis.
In view of the circumstances mentioned above, it is an object of the present invention to provide a fringe analysis error detection method utilizing a Fourier transform fringe analyzing method which can favorably detect, without complicating the apparatus configuration when analyzing fringe image data obtained by use of the phase shift method, influences of errors in the amount of displacement of phase shift and/or in the amount of relative inclination between the object to be observed and the reference.
It is another object of the present invention to provide a fringe analysis error correction method utilizing a Fourier transform fringe analyzing method which can favorably correct, without complicating the apparatus configuration when analyzing fringe image data obtained by use of the phase shift method, influences of errors in the amount of displacement of phase shift and/or in the amount of relative inclination between the object to be observed and the reference.
The present invention provides a fringe analysis error detection method in which phase shift elements are used for relatively phase-shifting an object to be observed and a reference with respect to each other, and a wavefront of the object is determined by fringe analysis;
the method comprising the steps of Fourier-transforming two pieces of carrier fringe image data respectively carrying wavefront information items of the object before and after the phase shift; and carrying out a calculation according to a result of the transform so as to detect an amount of error of the phase shift.
Here, the xe2x80x9camount of error of the phase shiftxe2x80x9d includes at least xe2x80x9camount of relative inclination between the object to be observed and the referencexe2x80x9d and xe2x80x9camount of translational displacement of the phase shift.xe2x80x9d
In the fringe analysis error detection method in accordance with the present invention, the fringe image data may be carrier fringe image data on which carrier fringes are superposed.
In the fringe analysis error detection method in accordance with the present invention, the two pieces of carrier fringe image data may be Fourier-transformed so as to determine carrier frequencies, and a position of a spectrum may be calculated according to two of the carrier frequencies so as to detect an amount of relative inclination between the object and reference generated by the phase shift.
When determining the carrier frequencies of carrier fringes in this case, a positional coordinate of a predetermined peak among peaks on a frequency coordinate system obtained by the Fourier transform may be determined, and an arithmetic operation for calculating the carrier frequencies may be carried out according to the positional coordinate.
In the fringe analysis error detection method in accordance with the present invention, the two pieces of carrier fringe image data may be Fourier-transformed so as to determine phase information of the object, and
thus obtained phase information of the object may be subjected to a predetermined arithmetic operation so as to detect an inclination of the object.
When determining the phase information of the object in this case, a predetermined spectrum distribution among spectrum distributions on a frequency coordinate system obtained by the Fourier transform may be determined, and an arithmetic operation for calculating the phase information according to the spectrum distribution may be carried out.
The predetermined arithmetic operation may be an arithmetic operation for determining a least square of the phase information of the object.
In the fringe analysis error detection method in accordance with the present invention, the two pieces of carrier fringe image data may be Fourier-transformed so as to determine complex amplitudes of carrier fringes, and a position may be calculated according to two of the complex amplitudes so as to detect the amount of translational displacement of the phase shift.
In the fringe analysis error detection method in accordance with the present invention, the two pieces of carrier fringe image data may be Fourier-transformed so as to determine carrier frequencies and complex amplitudes, and a position may be calculated according to two of the carrier frequencies and two of the complex amplitudes so as to detect the amount of relative inclination between the object and reference and the amount of tilt displacement of the phase shift which are generated by the phase shift.
In the fringe analysis error detection method in accordance with the present invention, the fringe images may be an interference fringe images.
Also, the present invention provides a fringe analysis error correction method in which, after the detection is carried out in the fringe analysis error detection method in accordance with the present invention, a correction calculation for compensating for the difference of the detected amount of inclination generated by the phase shift from a target amount of inclination and/or the difference of a predetermined amount of translational displacement of phase shift from a target predetermined amount of displacement of phase shift is carried out in the fringe analysis of the fringe image data.
The above-mentioned methods in accordance with the present invention are applicable to fringe image analyzing techniques using the Fourier transform method in general, such as analysis of interference fringes and moirxc3xa9 fringes, three-dimensional projectors based on fringe projection, or the like, for example.